Ultimate Limit State Design of Ship Structures(1)

Ultimate Limit State Design of Ship Structures

2003 8

2003 6

Owen F. Hughes

Acknowledgements

This dissertation is submitted as partial fulfillment of the requirements for the

degree of Doctor of Philosophy at the Pusan National University. My sincere

appreciation goes to my supervisor, Professor Jeom Kee Paik. His guidance, patience,

and encouragement were invaluable to the progress and completion of this study. This

study was supported by the American Bureau of Shipping, the Brain Korea 21 project and Pusan National University who are thanked for their assistance.

I deeply thank the co-authors of several papers published in SNAME Transactions,

including Dr. Anil Kumar Thayamballi, Dr. Ge Wang, Dr. Yung Sup Shin and Dr.

Donald Liu, for their academical discussions and allowance to reprint them.

I am grateful to all my committee members, Professor Sung Won Kang

(Committee Chairman), Professor Owen F. Hughes, Professor Jae Yong Ko and Professor Jae Myung Lee, for their helpful discussions and spent time on my behalf. I

also thank all fellow students in Ship Structural Mechanics Lab. for their help and the

great time we shared together.

Last but not least, I am indebted to my parents, my wife and her parents for love,

support and encouragement. I would like to dedicate this dissertation to my wife, Joo

Hyun Chun, and our expected child.

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Contents

Nomenclature i v

List of Tables viii

List of Figures x

1. Introduction 1

2. Ultimate Limit State Design of Ship Plating 6

2.1 Buckling / Ultimate Strength Design Procedure 13

2.2 Geometric and Material Properties 14

2.3 Load (Stress) Application 15 2.4 Modeling of Fabrication Related Imperfections 17

2.5 Buckling Based Capacity 22

2.5.1 Design Equations 22

2.5.2 Validity of the Johnson-Ostenfeld Equation 24

2.5.3 Effect of Rotational Restraints 26

2.5.4 Effect of Residual Stresses 35

2.5.5 Effect of Lateral Pressure 39

2.5.7 Effect of In-plane Bending 42

2.5.8 Elastic Edge Shear Buckling 44

2.6 Ultimate Strength Based Capacity 44

2.6.1 Ultimate Strength Equation for Combined Longitudinal Axial Load and

Lateral Pressure 45

2.6.2 Ultimate Strength Equation for Combined Transverse Axial Load and Lateral

Pressure 52

2.6.3 Ultimate Strength Equation for Edge Shear 54

2.6.4 Ultimate Strength Equation for Combined Biaxial Load, Edge Shear and

Lateral Pressure 54

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2.7 Comparison between Buckling and Ultimate Strength Based Capacities 59

3. Ultimate Limit State Design of Ship Stiffened Panels and Grillages 63

3.1 Modeling of Ship Stiffened Plate Structure 75

3.1.1 Panel Geometry 75

3.1.2 Panel Material Properties 76

3.1.3 Panel Boundary Conditions 77

3.1.4 Load Effects 78

3.1.5 Fabrication Related Initial Imperfections 80

3.2 Ultimate Strength Formulations for Collapse Mode I 82

3.2.1 Combined Longitudinal Axial Stress and Lateral Pressure 83

3.2.2 Combined Transverse Axial Stress and Lateral Pressure 85

3.2.3 Combined Edge Shear Stress and Lateral Pressure 86

3.2.4 Combined Biaxial Stresses, Edge Shear Stress and Lateral Pressure 88

3.3 Ultimate Strength Formulations for Collapse Mode II 89




3.3.4 Combined Biaxial Stress, Edge Shear Stress and Lateral Pressure 91

3.4 Ultimate Strength Formulations for Collapse Mode III 91



3.4.3Combined Edge Shear Stress and Lateral Pressure 102


3.5 Ultimate Strength Formulations for Collapse Mode IV 102




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3.6 Ultimate Strength Formulations for Collapse Mode V 108





3.7 Ultimate Strength Formulation for Collapse Mode VI 117

3.8 Verification Examples and Discussion 117

3.8.1 Ultimate Strength Characteristics of Longitudinally Stiffened Panels in Ships

118

3.8.2 Ultimate Strength Characteristics of Ship Grillages Comparisons with the

Smith Tests 131

4. Ultimate Limit State Design of Ship Hulls 137

4.1 Efficient and Accurate Methodology for the Progressive Collapse Analysis of

Ships 137

4.2 Progressive Collapse Characteristics of Typical Merchant Ships 147

4.2.1 Progressive collapse behavior under vertical moment 153

4.2.2 Effect of lateral pressure 165

4.2.3 Effect of horizontal moment 169

4.3 Closed-Form Ultimate Strength Formulation for Ship Hulls 172

4.4 Ultimate Limit State Design Format 188

4.5 Assessment of Safety Measure 192

5. Concluding Remarks 199

References 202

Appendix 213

Abstract (Korean) 229 Published Papers Related to This Study 231

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Nomenclature

Geometric Properties

a = length of plating, spacing between two adjacent transverses (frames) xA , yA = cross-sectional areas of a single x - or y -stiffener with attached effective

plating

ea = effective length of the plating corresponding to length a

eua = ea at plate ultimate limit state

sxA , syA = cross-sectional areas of a single x - or y -stiffener

b = breadth of plating, spacing between two adjacent longitudinals B = breadth of the stiffened panel, spacing between two adjacent longitudinal girders

eb = effective width of the plating corresponding to breadth b

eub = eb at plate ultimate limit state

fxb , fyb = breadths of x - or y -stiffener flange

wxh , wyh = heights of x - or y -stiffener web

xI , yI = moments of inertia of x - or y -stiffener with attached effective plating

fxJ , fyJ = torsion constants of x - or y -stiffener flange

wxJ , wyJ = torsion constants of x - or y -stiffener web

k = radius of gyration of longitudinals with attached effective plating

(= xx A/I ) L = length of the stiffened panel, spacing between two adjacent transverse frames (or bulkheads)

sxn , syn = numbers of x - or y -stiffeners

t = thickness of plating

eqt = equivalent plate thickness

fxt , fyt = thicknesses of x - or y -stiffener flange

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wxt , wyt = thicknesses of x - or y -stiffener web

xZ , yZ = section moduli of x - or y -stiffener with attached effective plating

oxz , oyz = distances from the middle plane of the plating to the neutral axis of x - or

y -stiffener with attached full plating

pxz , pyz = distances from the middle plane of the plating to the neutral axis of x - or

y -stiffener with attached effective plating

fxz , fyz = distances from the middle plane of the stiffener flange to the neutral axis of

x - or y -stiffener with attached effective plating

= reduced slenderness ratio of the plating between longitudinals ( )E/t/b o= = slenderness ratio of longitudinals with attached effective plating

( E/k/a ox= )Eox / =

Material Properties

D = bending rigidity of isotropic plating ( ))1(12/Et 23 = E = Youngs modulus

G = shear modulus ( ))1(2/E += = Poissons ratio

o = yield stress for plating ( op= )

oeq = equivalent yield stress for the entire stiffened panel

oyox , = equivalent yield stresses for the stiffened panel in the x or y direction

os = yield stress of stiffeners

o = shear yield stress of plating ( 3/o= )

Initial Imperfections

omA = initial deflection amplitude with respect to buckling half wave number m

in the x direction

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onA = initial deflection amplitude with respect to buckling half wave number n in

the y direction

ta , tb = breadths of tensile residual stress block for rty or rtx

oplw = maximum initial deflection of plating between stiffeners

osxw , osyw = column type initial deflections of x - or y -stiffener

rcx , rcy = compressive residual stresses of plating between stiffeners in the x or

y direction

rsx , rsy = compressive residual stresses in the x - or y -stiffener web

rtx , rty = tensile residual stresses of plating between stiffeners in the x or y

direction

Applied Loads

p = net lateral pressure on panel

q = lateral line load on plate-stiffener combination

1x , 2x = minimum or maximum axial stresses in the x direction

xav , yav = average axial stresses in the x or y direction

xM , yM = axial stresses at the most highly stressed x - or y -stiffener

1y , 2y = minimum or maximum axial stresses in the y direction

av = average edge shear stress

Buckling and Ultimate Strength

uop = ultimate strength under p alone

cr = critical buckling stress

E = elastic buckling stress

u = ultimate stress

xu , yu = ultimate longitudinal or transverse axial strength components

E = elastic shear buckling strength

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u = ultimate shear strength

uo = ultimate strength under av alone

Others

m , n = buckling half wave numbers in the x or y direction

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List of Tables

Table 2.1 Initial deflection amplitudes for various initial deflection shapes indicated in

Fig.2.7

Table 3.1 Mean values of geometric and material properties for the Tanaka & Endo test

structures

Table 3.2 Geometric characteristics of the Tanaka & Endo test structures

Table 3.3 Initial imperfections for plating and longitudinals in the Tanaka & Endo test

structures

Table 3.4 Comparison of the ALPS/ULSAP with the Tanaka & Endo experiment and

FEA

Table 3.5 Mean values of geometric properties and material yield stresses for the Smith

test grillages

Table 3.6 Other Geometric characteristics of the Smith test grillages

Table 3.7 Initial imperfections of plating, longitudinals and transverses for the Smith

test grillages

Table 3.8(a) Comparison of the Smith FEA with the experiment for ultimate strength of grillages

Table 3.8(b) Comparison of ALPS/ULSAP with the Smith experiments and FEA for ultimate strength of grillages

Table 4.1 Hull sectional properties of the 10 typical merchant ships

Table 4.2 A comparison of the ultimate hull girder strength calculations obtained by the

ALPS/HULL and the closed-form design formula (DF) for 10 typical commercial ships indicated in Fig.4.9

Table 4.3(a) Hull sectional properties of the existing double hull tankers Table 4.3(b) The computed ultimate hull girder strength of the existing double hull

tankers

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Table 4.4(a) Hull sectional properties of the existing bulk carriers Table 4.4(b) The computed ultimate hull girder strength of the existing bulk carriers Table 4.5(a) Hull sectional properties of the existing container vessels Table 4.5(b) The computed ultimate hull girder strength of the existing container

vessels

Table 4.6 Safety measure calculations for the 10 typical merchant ships

Table 4.7 Safety measure calculations for the 9 existing double hull tankers

Table 4.8 Safety measure calculations for the 12 existing bulk carrier

Table 4.9 Safety measure calculations for the 9 existing container vessels

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List of Figures

Fig.1.1. Structural design considerations based on the ultimate limit state

Fig.2.1. A schematic of the collapse behavior of steel plating under predominantly

compressive loads

Fig.2.2. A typical stiffened plate structure in a ship

Fig.2.3. Typical geometry for the longitudinals and transverses

Fig.2.4. The plating under a general pattern of combined external loads

Fig.2.5. Idealized load application for the plating under uniform biaxial, edge shear and

lateral pressure loads

Fig.2.6. Fabrication related initial deflections in steel stiffened panels

Fig.2.7. Some typical patterns of welding induced initial deflection in ship plating

Fig.2.8. Idealization of welding induced residual stress distribution inside plating in the

x and y directions

Fig.2.9. The ultimate capacity versus the elastic bifurcation buckling stress of plating

under longitudinal compression alone, 0.3b/a =


under transverse compression along, 3b/a =


under edge shear alone, 3b/a =

Fig.2.12. Buckling coefficient 1xk for a plate under longitudinal compression,

elastically restrained at the long edges and simply supported at the short edges as

obtained by directly solving the buckling characteristic equation and by the

proposed approximate equation


elastically restrained at the short edges and simply supported at the long edges as

obtained by directly solving the buckling characteristic equation

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Fig.2.14. Buckling coefficient 1yk for a plate under transverse compression, elastically

restrained at the long edges and simply supported at the short edges as obtained by

directly solving the buckling characteristic equation

Fig.2.15. Buckling coefficient 2yk for a plate under transverse compression,



Fig.2.16. Accuracy of the design equation for the buckling coefficient 2xk

Fig.2.17. Accuracy of the design equation for the buckling coefficient 1yk


Fig.2.19. Elastic buckling interaction relationships for plating under combined biaxial

compression

Fig.2.20. Elastic buckling interaction relationships of plating under combined axial

compression and edge shear

Fig.2.21(a). Variation of the elastic compressive buckling stress as a function of the welding induced residual stress and the plate aspect ratio, 0rcy = , 50t/b = ,

07.2= , MPa352o = , rtx o8.0 = Fig.2.21(b). Variation of the elastic compressive buckling stress as a function of the

welding induced residual stress and the plate aspect ratio, 0rcy = , 100t/b = ,

14.4= , MPa352o = , rtx o8.0 = Fig.2.21(c). Variation of the elastic compressive buckling stress as a function of the

welding induced residual stress and the plate aspect ratio, rcxrcy a/b = ,

50t/b = , 07.2= , MPa352o = , ortx 8.0 = Fig.2.21(d). Variation of the elastic compressive buckling stress as a function of the

welding induced residual stress and the plate aspect ratio, rcxrcy a/b = ,

100t/b = , 14.4= , MPa352o = ortx 8.0 = Fig.2.22. Effect of lateral pressure on the plate compressive buckling strength

Fig.2.23. Membrane stress distribution inside the plate element under longitudinal

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compressive loads

Fig.2.24. Possible locations for the initial plastic yield at the plate edges under

combined uniaxial load and pressure

Fig.2.25. Comparison of the ALPS/ULSAP with mechanical test results and FEA for

long plating under uniaxial compression, reference numbers being extracted from

Ellinas et al.

Fig.2.26. Effect of initial deflection on the plate ultimate compressive strength

Fig.2.27(a). Comparison of the ALPS/ULSAP method with the Yamamoto collapse test results for plating under combined longitudinal axial compression and lateral

pressure loads, for 508.3= Fig.2.27(b). Comparison of the ALPS/ULSAP method with the Yamamoto collapse test

results for plating under combined longitudinal axial compression and lateral

pressure loads, for 554.2= Fig.2.27(c). Comparison of the ALPS/ULSAP method with the Yamamoto collapse test

results for plating under combined longitudinal axial compression and lateral

pressure loads, for 084.3= Fig.2.28. Variation of the ultimate transverse compressive strength of a long plating

shown as a function of the reduced slenderness ratio, 3b/a =

Fig.2.29. The ultimate strength versus the elastic bifurcation buckling stress of plating

under edge shear

Fig.2.30(a). Plate ultimate strength interaction between biaxial compression, 3b/a = , mm13t = , initial deflection shape #1

Fig.2.30(b). Plate ultimate strength interaction between biaxial compression, 3b/a = , mm21t = , initial deflection shape #1

Fig.2.31(a). Plate ultimate strength interaction between biaxial compression, 6b/a = , mm13t = , initial deflection shape #1

Fig.2.31(b). Plate ultimate strength interaction between biaxial compression, 6b/a = ,

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mm21t = , initial deflection shape #1

Fig.2.32(a). Plate ultimate strength interaction between axial compression and edge shear, 1b/a = and 3= , 2opl 1.0t/w =

Fig.2.32(b). Plate ultimate strength interaction between axial compression and edge shear, 1b/a = and 3= , 2opl 1.0t/w =

Fig.2.33(a). Plate capacity interactions between biaxial compression as those obtained by FEA, buckling and ultimate strength based capacity formulae,

,3b/a = mm13t = , initial deflection shape #1

Fig.2.33(b). Plate capacity interactions between biaxial compression as those obtained by FEA, buckling and ultimate strength based capacity formulae,


Fig.2.34(a). Plate capacity interactions between biaxial compression as those obtained by FEA, buckling and ultimate strength based capacity formulae,


Fig.2.34(b). Plate capacity interactions between biaxial compression as those obtained by FEA, buckling and ultimate strength based capacity formulae,


Fig.3.1. A ship grillage with support members in both directions

Fig.3.2. A cross-stiffened panel under combined in-plane and lateral pressure loads

Fig.3.3(a). Mode I-1: Overall collapse of a uniaxially stiffened panel Fig.3.3(b). Mode I-2: Overall collapse of a cross-stiffened panel Fig.3.3(c). Mode II: Plate induced failure - yielding at the corners of plating between

stiffeners

Fig.3.3(d). Mode III: Plate induced failure - yielding of plate-stiffener combination at mid-span

Fig.3.3(e). Mode IV: Stiffener induced failure - local buckling of the stiffener web Fig.3.3(f). Mode V: Stiffener induced failure - lateral-torsional buckling of stiffener

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Fig.3.4. A comparison of the ultimate strength formulations for plate-stiffener

combinations under axial compression

Fig.3.5. Typical cross-section types for longitudinals and transverses

Fig.3.6(a). An example pattern of post-weld initial deflections in steel stiffened panels Fig.3.6(b). Idealization of the distribution of welding induced residual stresses in the

plating between stiffeners in the x and y directions

Fig.3.7. Plasticity at panel longitudinal mid-edges for a combined xav and p

Fig.3.8. Plasticity at panel transverse mid-edges for a combined yav and p

Fig.3.9. Ultimate strength interaction relationship for a simply supported (isotropic) plate subjected to edge shear and lateral pressure

Fig.3.10. Variation of the effective width of a simply supported plate under uniaxial

compression

Fig.3.11. Variation of the ultimate compressive strength from the Perry-Robertson

formula versus the column slenderness ratio for plate-stiffener combinations,

under combined axial compression and lateral load

Fig.3.12. Example variation of elastic buckling coefficients for angle / T section

stiffener web with increase in aspect ratio and torsional rigidity of plating,

accounting for the influence of rotational restraints

Fig.3.13. General and assumed tripping deformations of a plate-stiffener combination

Fig.3.14. Effect of the ww t/h ratio on the tripping strength of a plate and flanged-

stiffener combination without considering the plate rotational restraints

Fig.3.15. The Tanaka & Endo test structure for longitudinally stiffened panels under

uniaxial compression, incorporating two dummy panels away from the transverse

frames

Fig.3.16. Correlation of ALPS/ULSAP predictions with experiments and FEA from

Tanaka & Endo

Fig.3.17. Extent taken for the ANSYS analysis with 1/2+1+1/2 bay model

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Fig.3.18. Boundary conditions adopted for the ANSYS analysis

Fig.3.19. ANSYS FE meshes together with boundary conditions

Fig.3.20(a). Node location in 2-2 section as defined in Fig.3.18 Fig.3.20(b). Node location in 3-3 section as defined in Fig.3.18 Fig.3.21(a). Comparisons of the ultimate strengths between ANSYS, ABAQUS,

ALPS/ULSAP and PULS for the panel with PULS default settings of initial

deflections, i.e., osxw =a/1000 = *osxw , oplw =b/200 (t=21mm) Fig.3.21(b). Comparisons of the ultimate strengths between ANSYS, ALPS/ULSAP

and PULS for the panel with PULS default settings of initial deflections, i.e.,

osxw =a/1000 = *osxw , oplw =b/200 (t=21mm) Fig.3.21(c). Comparisons of the ultimate strengths between ANSYS, ALPS/ULSAP


osxw =a/1000 = *osxw , oplw =b/200 (t=15mm) Fig.3.21(d). Comparisons of the ultimate strengths between ANSYS, ALPS/ULSAP


osxw =a/1000 = *osxw , oplw =b/200 (t=15mm) Fig.3.22. Correlation of the ALPS/ULSAP method with the experimental data and FE

solutions for the Smith test grillages

Fig.4.1. Various types of idealizations for a steel plated structure

Fig.4.2(a). The ISUM beam-column unit with attached plating Fig.4.2(b). The ISUM beam-column unit without attached plating Fig.4.3. The ISUM rectangular plate unit

Fig.4.4. The ISUM stiffened panel unit

Fig.4.5. Idealized stress-strain behavior of the ISUM plate or stiffened panel unit for the

ultimate strength analysis

Fig.4.6(a). Mid-ship section of the Dow frigate test ship Fig.4.6(b). ALPS/HULL model for the Dow frigate test hull

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Fig.4.6(c). Comparison of ALPS/HULL with the Dow test results, varying the level of initial imperfections

Fig.4.7(a). ALPS/HULL model I Fig.4.7(b). ALPS/HULL model II Fig.4.7(c). ALPS/HULL model III Fig.4.7(d). ALPS/HULL model IV Fig.4.7(e). ALPS/HULL model V Fig.4.7(f). ALPS/HULL model VI Fig.4.8. Progressive collapse behavior of a 105,000 DWT double hull tanker hull with

one center-longitudinal bulkhead under vertical bending moment, as obtained by

the six types of modeling methods

Fig.4.9(a). Schematic representation of mid-ship section of a 254,000 DWT single hull tanker

Fig.4.9(b). Schematic representation of mid-ship section of a 105,000 DWT double hull tanker with one center-longitudinal bulkhead

Fig.4.9(c). Schematic representation of mid-ship section of a 313,000 DWT double hull tanker with two longitudinal bulkheads

Fig.4.9(d). Schematic representation of mid-ship section of a 170,000 DWT single sided bulk carrier

Fig.4.9(e). Schematic representation of mid-ship section of a 169,000 DWT double sided bulk carrier

Fig.4.9(f). Schematic representation of mid-ship section of a 3,500 TEU container vessel

Fig.4.9(g). Schematic representation of mid-ship section of a 5,500 TEU container vessel

Fig.4.9(h). Schematic representation of mid-ship section of a 9,000 TEU container vessel

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Fig.4.9(i). Schematic representation of mid-ship section of a 113,000 DWT FPSO (floating, production, storage and offloading unit)

Fig.4.9(j). Schematic representation of mid-ship section of a 165,000 DWT shuttle tanker

Fig.4.10(a). Progressive collapse behavior of the 254,000 DWT single hull tanker under vertical moment varying the level of initial imperfections, as obtained by

ALPS/HULL

Fig.4.10(b). Progressive collapse behavior of the 105,000 DWT double hull tanker with one center-longitudinal bulkhead under vertical moment varying the level of initial

imperfections, as obtained by ALPS/HULL

Fig.4.10(c). Progressive collapse behavior of the 313,000 DWT double hull tanker with two longitudinal bulkheads under vertical moment varying the level of initial

imperfections, as obtained by ALPS/HULL

Fig.4.10(d). Progressive collapse behavior of the 170,000 DWT single sided bulk carrier under vertical moment varying the level of initial imperfections, as

obtained by ALPS/HULL

Fig.4.10(e). Progressive collapse behavior of the 169,000 DWT double sided bulk carrier under vertical moment varying the level of initial imperfections, as


Fig.4.10(f). Progressive collapse behavior of the 3,500 TEU container vessel under vertical moment varying the level of initial imperfections, as obtained by

ALPS/HULL

Fig.4.10(g). Progressive collapse behavior of the 5,500 TEU container vessel under vertical moment varying the level of initial imperfections, as obtained by

ALPS/HULL

Fig.4.10(h). Progressive collapse behavior of the 9,000 TEU container vessel under vertical moment varying the level of initial imperfections, as obtained by

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ALPS/HULL

Fig.4.10(i). Progressive collapse behavior of the FPSO hull under vertical moment varying the level of initial imperfections, as obtained by ALPS/HULL

Fig.4.10(j). Progressive collapse behavior of the shuttle tanker hull under vertical moment varying the level of initial imperfections, as obtained by ALPS/HULL

Fig.4.11. Variation of the neutral axis due to structural failure for the single hull tanker,

as obtained by ALPS/HULL

Fig.4.12(a). Schematic of water pressure distribution for the 313,000 DWT double hull tanker with two longitudinal bulkheads, being a sum of static and hydrodynamic

pressure for head sea state

Fig.4.12(b). Schematic of water pressure distribution for the 170,000 DWT single sided bulk carrier, being a sum of static and hydrodynamic pressure for head sea state

Fig.4.12(c). Schematic of water pressure distribution for the 9,000 TEU container vessel, being a sum of static and hydrodynamic pressure for head sea state

Fig.4.13(a). Progressive collapse behavior of the 313,000 DWT double hull tanker with two longitudinal bulkheads under vertical moment varying the magnitude of water

pressure, as obtained by ALPS/HULL

Fig.4.13(b). Progressive collapse behavior of the 170,000 DWT single sided bulk carrier under vertical moment varying the magnitude of water pressure, as


Fig.4.13(c). Progressive collapse behavior of the 9,000 TEU container vessel under vertical moment varying the magnitude of water pressure, as obtained by

ALPS/HULL

Fig.4.13(d). Variation of the ultimate hull girder strengths as a function of the magnitude of water pressure, as obtained by ALPS/HULL

Fig.4.14(a). Progressive collapse behavior of the 313,000 DWT double hull tanker with two longitudinal bulkheads under combined vertical and horizontal moments, as

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Fig.4.14(b). Progressive collapse behavior of the 170,000 DWT single sided bulk carrier under combined vertical and horizontal moments, as obtained by

ALPS/HULL

Fig.4.14(c). Progressive collapse behavior of the 9,000 TEU container vessel under combined vertical and horizontal moments, as obtained by ALPS/HULL

Fig.4.14(d). Ultimate hull girder strength interaction relationships between vertical and horizontal moments, as obtained by ALPS/HULL

Fig.4.15. Variation of the longitudinal stress distribution during the progressive collapse

under hogging moment, as obtained by ALPS/HULL (a) pre-ultimate limit regime (b) ultimate limit state

Fig.4.16. Longitudinal stress distribution over a ships cross-section at the overall

collapse state as suggested by Paik & Mansour (1995) Fig.4.17(a). Correlation between ALPS/HULL progressive collapse analyses and the

closed-form design formula predictions for a slight level of initial imperfections

for 10 typical commercial ships indicated in Fig.4.9

Fig.4.17(b). Correlation between ALPS/HULL progressive collapse analyses and the closed-form design formula predictions for an average level of initial

imperfections for 10 typical commercial ships indicated in Fig.4.9

Fig.4.17(c). Correlation between ALPS/HULL progressive collapse analyses and the closed-form design formula predictions varying the level of initial imperfections

for 10 typical commercial ships indicated in Fig.4.9

Fig.4.18. Correlation between ALPS/HULL progressive collapse analyses and the

closed-form design formula predictions for the existing double hull tankers


closed-form design formula predictions for the existing bulk carriers


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closed-form design formula predictions for the existing container vessels


closed-form design formula predictions for all (30) existing vessels considered Fig.4.22 The section modulus based safety measure versus the ultimate strength based

safety measure for the 10 typical merchant ships

Fig.4.23 The section modulus based safety measure versus the ultimate strength based

safety measure for all (30) existing vessels Fig.A.1. A schematic of the total membrane stress distribution inside the plating in the

x direction

Fig.A.2 A schematic of the total membrane stress distribution inside the plating in the

y direction

Fig.A.3. Geometric properties of longitudinals or transverses with attached effective

plating

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1. Introduction

Ship structures while in service are likely subjected to various types of loads and deformations that may be range from the routine to the extreme or accidental. The

structure is designed so that it should sustain such loads and deformations throughout

its lifetime.

Two types of structural design methods are relevant, namely allowable stress

design (ASD) and limit state design (LSD). In the former, the design is undertaken so that the stresses resulting from the design loads should be kept under a certain working

stress level which is usually determined based upon past experience. In the latter, the

design is based on the limit state which represents a condition that the structure fails to

fulfill its intended function.

It is now well recognized that the LSD is a better basis for structural design

because it is much more effective to determine the real safety measures of any structure.

In this regard, the design of land-based structures and naval vessel structures has been

undertaken based on the LSD, while the structural design of merchant ships is still

being performed following the traditional ASD together with buckling strength check. It

is however clear that the basis of structural design is now moving from the ASD to the

LSD.

The LSD is normally categorized into the following four types, namely (Paik & Thayamballi 2003)

Serviceability limit state design

Ultimate limit state design

Fatigue limit state design

Accidental limit state design

It is noted that these various types of LSD may need to be considered depending

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upon the conditions or situations of loading or structure types, among others. Also, the

methodologies and the safety level of individual LSD are different as well. The present

dissertation is concerned with the ultimate limit state design of ship structures that

involves plastic collapse or ultimate strength.

Proportional limit

Buckling strength

Ultimate strengthLinearelasticresponse

Displacement

Forc

e

Design load level

A

B

B*

Design load level

2

1

Fig.1.1. Structural design considerations based on the ultimate limit state (Paik & Thayamballi 2003)

Figure 1.1 shows a schematic representation of a structure under predominantly

axial compressive loads. The current ship structural design method uses the elastic

buckling strength with a simple plasticity correction which is represented by the point A

in Fig.1.1. In this case, the structural designer does not have any information in the

post-buckling regime.

When the load level 2 is applied as indicated in Fig.1.1, the structure will be safe,

but if the load level 1 is applied the structure will possibly collapse. Even though the

ultimate strength represented by point B would be expected higher than the point A, it is

not possible to determine the real safety margin against the ultimate strength which is

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represented by point B as long as point B remains unknown. The primary aim of the

present study is to develop the efficient and accurate methodologies of determining

point B, i.e., the ultimate strength of ship structures.

The dissertation is composed of 5 chapters. Following chapter 1 Introduction,

chapter 2 presents the ultimate strength design methodology of plates within stiffened

panels. The behavior of ship plating normally depends on a variety of influential factors,

namely geometric / material properties, loading characteristics, initial imperfections,

boundary condition. To achieve a more advanced buckling and ultimate strength design

of ship plating, this chapter focuses on the following three subjects which have been studied theoretically, numerically and experimentally:

Modeling of post-weld initial imperfections (i.e., initial deflections and residual stresses) and their effects,

Influence of rotational restraints and torsional rigidity of support members on the

plate buckling strength, and

Ultimate strength design equations under combined loads including biaxial

compression / tension, edge shear and lateral pressure loads.

Chapter 3 aims to deal with the advanced ultimate strength design of stiffened

panels and grillages that form parts of ship structures. In contrast, the chapter 2 dealt

with the ultimate strength design of plate elements within such stiffened panels. To

achieving a more advanced ultimate strength design of stiffened panels and grillages,

this chapter proposes a more advanced design oriented ultimate strength design

methodology for ship stiffened panels and grillages. Possible failure modes involved in

collapse of stiffened panels and grillages are six as categorized by Paik et al. (2001c) and Paik & Thayamballi (2003). The ultimate strength of the stiffened panel under combined loads is calculated taking into account all of the possible failure modes and

the interplay of various factors such as geometric and material properties, loading and

-4-

post-weld initial imperfections. As is usual, the collapse of stiffened panels is

considered to occur at the lowest value among the various ultimate loads calculated for

each of the collapse patterns. The design oriented strength formulations developed

accommodate all potential applied load components including biaxial compression /

tension, biaxial in-plane bending, edge shear and lateral pressure loads. The fabrication

related initial imperfections (initial deflections and residual stresses) are included in the developed strength formulations as parameters of influence. The validity of the

proposed ultimate strength formulations is confirmed by a comparison with the

nonlinear finite element solutions and mechanical collapse test results. Important

insights developed from the present study are summarized. Predictions from the design

oriented procedures proposed in chapter 2 and 3 are automated using a computer

program called ALPS/ULSAP which stands for Analysis of Large Plated Structures /

ULtimate Strength Analysis of Panels, developed by Professor Paik (Paik & Thayamballi 2003).

The subject of chapter 4 is to deal with the calculation of the ultimate hull girder strength using the methodology for ultimate strength of ship structures developed in

chapter 2 and chapter 3. A special purpose program, ALPS/HULL, developed by

Professor Paik (Paik & Thayamballi 2003), for the automated calculation of the progressive collapse analysis of ships hulls under extreme hull girder loads is used,

which can include any combination of vertical bending, horizontal bending, sectional

shear and torsion. The characteristics of progressive collapse behavior for a total of 10

typical merchant ships under vertical bending moment are investigated using the

developed ultimate strength design formulations. Effects of lateral pressure and

horizontal moment on the hull girder ultimate vertical moment are also studied.

Through the above insight closed-form ultimate strength formulations for the ultimate

strength of ships under vertical bending moment are developed. Ultimate strength of the

hull girders for a total of 40 merchant ships obtained by ALPS/HULL and the closed-

-5-

form design formula are compared to verify the applicability of closed-form design

formula. The ultimate limit state design format for ships is addressed. Finally, The

section modulus based safety measure and the ultimate strength based safety measure

for all (40) target vessels are compared. The developed closed-form design formula is programmed into ALPS/USAS-S (Paik & Thayamballi 2003) to predict the ultimate hull girder strength of ships under vertical bending moments effectively. For more

information about computer programs mentioned above see the reference or visit

http://ssml.naoe.pusan.ac.kr.

-6-

2. Ultimate Limit State Design of Ship Plating

Most of contents in this chapter are reprinted from the paper of SNAME

Transactions (Paik et al. 2000b), which dealt with the advanced buckling and ultimate strength design of ship plates within stiffened panels, where the author was involved as

a coauthor of the paper. In this regard, the author is pleased to acknowledge that the rest

of the paper authors allowed him to reprint the results here.

The overall failure of a ship hull girder is normally governed by buckling and

plastic collapse of the deck, bottom or sometimes the side shell stiffened panels.

Therefore, the relatively accurate calculation of buckling and plastic collapse strength

of stiffened plating of the deck, bottom and side shells is a basic requirement for the

safety assessment of ship structures. In stiffened panels, local buckling and collapse of

plating between stiffeners is a primary failure mode, and thus it would also be important

to evaluate the buckling and collapse strength interactions of plating between stiffeners

under combined loading.

The behavior of ship plating normally depends on a variety of influential factors,

namely geometric/material properties, loading characteristics, initial imperfections (i.e., initial deflections and residual stresses), boundary conditions and existing local damage related to corrosion, fatigue crack and denting.

The geometry of plating found in ship and offshore structures is normally

rectangular and the material used is usually mild or high tensile steel (Note that the use of aluminum alloys is increasing in the design and fabrication of high speed vessel

structures). The boundary condition for the rectangular plate elements making up steel plated structures is normally assumed to be simply supported or sometimes clamped for

practical purposes of analysis. In real ship plating, however, such ideal edge conditions

may never occur due to rotational restraint by support members along the plate edges.

The ship plating is generally subjected to combined in-plane and lateral pressure

-7-

loads. In-plane loads include biaxial compression / tension and edge shear, which are

mainly induced by overall hull girder bending and / or torsion of the vessel. Lateral

pressure loads are due to water pressure and / or cargo. The extreme of such load

components may not occur simultaneously, and more than one load component may

normally exist and interact. Hence, for more advanced design of ship structures, it is of

crucial importance to better understand the characteristics of the buckling and ultimate

strength for ship plating under combined loads.

Elastic BifurcationUltimate Strength

Perfect Thin PlatePerfect Thick PlateImperfect Plate

Axial Compressive Strain

Axia

l Com

pres

sive St

ress

Elastic BifurcationUltimate StrengthElastic BifurcationUltimate Strength

Perfect Thin PlatePerfect Thick PlateImperfect Plate

Axial Compressive Strain

Axia

l Com

pres

sive St

ress

Fig.2.1. A schematic of the collapse behavior of steel plating under predominantly

compressive loads

Since the post-weld initial imperfections in the form of initial deflections and

residual stresses exist in ship steel plating and can affect significantly the strength, such

welding induced initial imperfections should be included in the strength calculations as

parameters of influence.

When a perfectly flat plate (i.e., without initial imperfections) is subjected to predominantly compressive loads, buckling (bifurcation) can occur if the applied

-8-

compressive stress reaches a critical bifurcation stress, see Fig.2.1. However, the in-

plane stiffness of plating with initial imperfections decreases from the very beginning

as the compressive loads increase. In this more general case, it is not possible to define

a bifurcation point for buckling.

The phenomenon of buckling may be categorized by plasticity considerations into

three classes, namely elastic buckling, elastic-plastic buckling and plastic buckling, the

last two being called inelastic buckling (Paik & Thayamballi 2003). The first class (i.e., elastic buckling) typically indicates that buckling occurs solely in the elastic regime. This class of buckling is often seen in very thin steel plates. The second (i.e., elastic-plastic buckling) normally represents the case wherein buckling occurs after plastification has occurred in a local region in the plate. The third (i.e., plastic buckling) indicates that buckling occurs in the regime of gross yielding, i.e., after the plate has

yielded over large areas. Relatively thick plating may exhibit either elastic-plastic or

plastic buckling.

Unlike columns, plating can normally sustain additional applied loads even after

elastic buckling occurs since membrane tension develops along the plate edges resists

any abrupt increase in lateral deflection. A plate buckled in the elastic regime will

eventually collapse by a rapid decrease of in-plane stiffness (or an abrupt increase of lateral deflection) as the yield zone inside the plate is expanded. On the other hand, if buckling occurs in the elastic-plastic or plastic regime the plating normally immediately

reaches the ultimate limit state.

From the viewpoint of a structural designer, it can be said with reasonable

certainty that the buckling and ultimate strength problem for ship plating under a single

load application and common idealized edge conditions (e.g., simply supported along four edges) has been almost completely solved. In the more general case, however, we are still confronted with a number of problem areas that remain unsolved due to the

various influential factors previously mentioned. In the following, a literature review

-9-

of selected studies related to the buckling and ultimate strength of plating is now made.

Depending on the rotational restraints and torsional rigidity of support members

along the plate edges, the common ideal edge conditions (i.e., the assumption that the plate edges are simply supported or clamped) may or may not be appropriate to apply (Bleich 1952, Timoshenko & Gere 1963). The plate element is normally subjected to combined loads and the buckling mode depends on the interaction of these load

components. Therefore, the plate buckling strength should in principle be evaluated by

taking into account the effects of boundary condition and load component interactions

among other factors.

Williams (1976) investigated the buckling strength characteristics of plate elements varying torsional rigidity of support members along their edges. Paik et al.

(1993) surveyed the bending and torsional rigidities of support members for plate elements in merchant vessel structures. Based on the survey results, they concluded that

due to the rotational restraint by support members at plate edges the plate edge

condition would be in an intermediate situation, i.e., between a simply supported and a

clamped condition. Most recently, Paik & Thayamballi (2000) investigated the buckling strength characteristics of steel plating elastically restrained at their edges and

developed simple design formulations for buckling strength as function of the torsional

rigidity of support members that provide the rotational restraints along either one set of

edges or all (four) edges. Mansour (1976) developed charts for predicting the buckling and post-buckling

behavior of simply supported plates under combined in-plane and lateral pressure loads.

Steen & Valsgard (1984) developed a simplified buckling and ultimate strength equation for plates under biaxial compression and lateral pressure loads. They define a

pseudo-buckling (non-bifurcation) strength for initially deflected plating. Ueda et al. (1985) developed elastic buckling interaction equations for simply supported plates subject to five load components, namely biaxial compression, biaxial in-plane bending

-10-

and edge shear. Paik et al. (1992a) developed the elastic buckling interaction equation for simply supported plates under biaxial compression, edge shear and lateral pressure

loads. The post-weld residual stresses were also later incorporated in the plate buckling

design formula (Paik et al. 1992b). To appropriately include the effects of post-weld initial imperfections in the strength calculations, an idealized model representing the

distribution of the post-weld initial imperfections is used. Mazzolani et al. (1998) studied the effect of welding on the local buckling of aluminum thin plates. The

influence of welding induced initial deflection and residual stresses on the buckling and

ultimate strength of plating under uniaxial compression and lateral pressure was studied

by Yao et al. (1998). Most design rules of classification societies approximately calculate the inelastic

buckling strength of plate elements by a correction for plasticity applied to the elastic

buckling strength, using the so-called Johnson-Ostenfeld formula. This approach

normally tends to underestimate the buckling strength for one single stress component

loading, but in some cases for combined loading it can overestimate the buckling

strength. Paik et al. (1992b) and Fujikubo et al. (1997) have derived newer empirical formulations of the plasticity correction by curve fitting based on nonlinear finite

element solutions.

Following von Karman et al. (1932), the concept of effective width has been recognized as an efficient device for characterizing the post-buckling strength behavior

of a plate in compression. For collapse strength prediction of steel plates, the effective

width concept has also been widely used (Faulkner 1975). For such use, the reduction of in-plane stiffness of a buckled plate is evaluated by using the effective width concept,

and it is assumed that the plate reaches the ultimate limit state if the normal stress

components within the plate field satisfy certain predefined ultimate strength criteria.

An extensive review of a number of studies for the derivation of the effective

width formulae for plates, undertaken until the early 80s, has been made by Rhodes

-11-

(1984). Since then, Ueda et al. (1986a) derived the effective width formula for a plate under combined biaxial compression and edge shear taking into account the effects of

initial deflections and welding induced residual stresses. Usami (1993) studied the effective width of plates buckled in compression and in-plane bending.

While the concept of effective width is aimed at the evaluation of in-plane stiffness

of plate elements buckled in compression, Paik (1995) suggested a new concept of the effective shear modulus to evaluate the effectiveness of plate elements buckled in edge

shear. The effective shear modulus concept is useful for computation of the post-

buckling behavior of plate girders under predominant shear forces.

Regarding the ultimate strength interaction equations for plate elements under

combined loads, a number of studies have also been undertaken in the past, e.g., for

uniaxial compression and shear (Fujita et al. 1979), for in-plane compression and tension (Smith et al. 1987), for uniaxial compression and lateral pressure (Aalami & Chapman 1972, Aalami et al. 1972, Okada et al. 1979, Paik & Kim 1988), for biaxial compression (Dier & Dowling 1983, Ohtsubo & Yoshida 1985), for biaxial compression and lateral pressure (Dowling & Dier 1978, Soreide & Czujko 1983, Steen & Valsgard 1984, Davidson et al. 1991, Soares & Gordo 1996, Wang & Moan 1997), for biaxial compression and shear (Ueda et al. 1984, 1995, Davidson et al. 1989), for biaxial compression, shear and lateral pressure (Ueda et al. 1986b), among others. Some of the methods mentioned above approximately accommodate post-weld initial

imperfections, but others neglect them.

For safety assessment of aging ship structures, it is necessary to better understand

the influence of local damage related to corrosion, fatigue cracking and dents on the

strength. Smith & Dow (1981) review structural damage in a ships bottom or side shell as may be caused by collisions, grounding, hydrodynamic impact or explosions, with

particular reference to the influence of such damage on hull girder bending strength.

Paik et al. (1998a) proposed a probabilistic corrosion rate estimation model of ship

-12-

plating. They also studied the ultimate strength reliability of ship structures related to

corrosion damage (Paik et al. 1998b, 1998c). Mateus & Witz (1997, 1998) studied the buckling and post-buckling behavior of corroded steel plates using the nonlinear finite

element method.

Based on the literature surveys mentioned above, it is evident that some of the

issues that need to be studied further for facilitating more refined buckling and ultimate

strength of ship plating are as follows

Modeling of the fabrication related initial imperfections (i.e., initial deflections and residual stresses) and their effect,

Effects of rotational restraints and torsional rigidity of support members,

Ultimate strength interaction characteristics under any combination of potential

load components, including biaxial compression / tension, edge shear and lateral

pressure loads,

Effects of structural deterioration such as due to opening, corrosion, fatigue

cracking and local dents.

The chapter focuses on the following three subjects which are studied theoretically, numerically and experimentally:

Modeling of post-weld initial imperfections (i.e., initial deflections and residual stresses) and their effects,

Influence of rotational restraints and torsional rigidity of support members on the

plate buckling strength,

Ultimate strength design equations under combined loads including biaxial

compression / tension, edge shear and lateral pressure loads, and

Deterioration of ultimate strength arising from opening, corrosion, fatigue

cracking and local dents.

-13-

Selected useful results and insight developed are summarized, and

recommendations are made with respect to related enhancements in the advanced ship

structural design and also needed future research.

2.1 Buckling / Ultimate Strength Design Procedure

Stiffeners Plate field

Stiffened panel

Heavy longitudinals and transverses

Fig.2.2. A typical stiffened plate structure in a ship

Figure 2.2 shows a schematic of the typical steel plated structure. The response of

such a structure can be classified into three levels, namely the bare plate element level,

the stiffened panel level and the entire plated structure level. This chapter is concerned

with the design for the first level (i.e., the plating between longitudinals and transverses). In such a case, the structure is to be designed so that the capacity (resistance) with allowable usage factor should not be less than the corresponding applied loads. To prevent the structure from failure (instability) under applied loading, therefore, the following criterion is to be satisfied:

-14-

c

(2.1)

where = applied load (stress), c = structural capacity (stress), and = allowable usage factor which is the inverse of the conventional factor of safety.

The applied load (stress) components are to be determined using any acceptable method such as the finite element approach. The structural capacity is normally

determined based on either buckling or ultimate strength. This chapter focuses on the

advanced design equations for the capacity based on both buckling and ultimate

strength.

2.2 Geometric and Material Properties

(b) y-stiffener

bfytfy

hwyzoy

a

N. A.twy

t

(a) x-stiffener

bfxtfx

hwx

zox

b

N. A.twx

t

z z

y x

Fig.2.3. Typical geometry for the longitudinals and transverses

The length and breadth of plating are a and b , respectively. The long direction

is taken as the x axis and the short direction is taken as the y direction, that is,

1b/a . The thickness of plating is t . The Young modulus and Poisson ratio are E

and , respectively. The yield stress of material is o . The plating is supported by

longitudinals and transverses. Figure 2.3 shows a typical geometry of the supporting

members in the x and y directions. The rotational restraint parameters for the

-15-

boundary longitudinals and transverses are defined as follows

bDGJL

L = , aDGJS

S = (2.2)

where L , S = rotational restraint parameters for the longitudinals and transverses, respectively, with ( )+= 12

EG , 6

tbthJ3fxfx

3wxwx

L+

= , ( )23

112EtD

= ,

6tbth

J3fyfy

3wywy

S+

= .

For a simply supported condition, L and S are set to be zero, while their values will become infinity for a clamped edge condition. For practical purposes, the

value of the rotational restraint parameter for clamped edges may be considered to be

20.

2.3 Load (Stress) Application

Figure 2.4 shows a general loading condition on the plating between longitudinals

and transverses. For the plate capacity calculations, the distribution of applied loads is

often idealized by their average values, similar to that shown in Fig.2.5. The

compressive stress is taken as negative and the tensile stress is taken as positive. The

average values of the applied stresses (loads) are defined as follows

22x1x

xav

+= ,

22y1y

yav

+= , =av , 2

ppp 21 += (2.3)

where xav = average axial stress in the x direction, yav = average axial stress in the

y direction, av = average edge shear stress, and p = average net lateral pressure.

-16-

The effect of in-plane bending stress in the x or y direction is included in the

buckling based capacity analysis. The in-plane bending stresses are defined as follows

(For the symbols, see Fig.2.4)

b

a

x1

x2y1 y2

p1 p2

x

y

b

a

x1

x2y1 y2

p1 p2

x

y

Fig.2.4. The plating under a general pattern of combined external loads

av

y

x

p

a

b

av

xav

yav

av

y

x

p

a

b

av

xav

yav

Fig.2.5. Idealized load application for the plating under uniform biaxial, edge shear and

lateral pressure loads

-17-

==

==

=+

=

==

0if2

if

1,0if,11

1x2x

xav

1x2x2x1x

x1x1x

2xxxav

x

x

xav2xxav1xxb

(2.4a)

==

==

=+

=

==

0if2

if

1,0if,11

1y2y

yav

1y2y2y1y

y1y1y

2yyyav

y

y

yav2yyav1yyb

(2.4b)

where xb , yb = in-plane bending stress in the x or y direction, respectively.

For safety evaluation using Eq.2.1, the measure of the applied stresses can be

defined for combined loading, as follows

2av

2yav

2xav ++= (2.5)

2.4 Modeling of Fabrication Related Imperfections

To fabricate the ship stiffened plate structure, welding is normally used and thus

the post-weld initial imperfections (initial deflections and residual stresses) develop in the structure. In advanced ship structural design, capacity calculations of ship plating

should accommodate post-weld initial imperfections as parameters of influence. The

characteristics of the post-weld initial imperfections are uncertain, and an idealized

model is used.

-18-

B

Ly

xwosx

wopl

wopl

wosy

b

b

b

b

a aaa

Fig.2.6. Fabrication related initial deflections in steel stiffened panels

Figure 2.6 shows a schematic of the post-weld initial deflections in ship stiffened

plate structure. The measurements of welding induced initial deflection for plating in

merchant ship structures reveal a complex multi-wave shape in the long direction and

one half wave is found in the short direction (Carlsen & Czujko 1978, Antoniou 1980, Kmiecik et al. 1995). In this case, the plate initial deflection can approximately be expressed by

=

=

M

1ioi

opl

o

by

sina

xisinB

w

w (2.6)

where oplw = relative maximum initial deflection of the plating between stiffeners, and

oiB = initial deflection amplitudes normalized by oplw .

Paik & Pedersen (1996) examined 33 sets of measurements and showed that Eq.2.6 with 11M = could reasonably model the measured initial deflections. For the

shapes of initial deflection in ship plating shown in Fig.2.7, for instance, the

-19-

coefficients oiB are given as those indicated in Table 2.1. Smith et al. (1987) suggest the following maximum values of representative initial deflections for plating in

merchant vessel structures which may be used to approximate oplw in Eq.2.6:

=

levelseriousfor3.0levelaveragefor1.0

levelslightfor025.0

t

w

2

2

2

opl

(2.7)

0

1

wo / w

opl

a/2 a

(a) Initial deflection shape #1

0

1

wo / w

opl

a/2 a

(b) Initial deflection shape #2

0

1

wo / w

opl

a/2 a

(c) Initial deflection shape #3

a/2 a

wo / w

opl

0

1

(d) Initial deflection shape # 4 Fig.2.7. Some typical patterns of welding induced initial deflection in ship plating

-20-

rcy

rty

rcx

rtx

Comp.

Tens.Tens.

x

yat ata 2at

b2b

tb t

b t

Fig.2.8. Idealization of welding induced residual stress distribution inside plating in the

x and y directions

Table 2.1 Initial deflection amplitudes for various initial deflection shapes indicated in

Fig.2.7 Initial

Deflection Shape No.

1oB 2oB 3oB 4oB 5oB 6oB 7oB 8oB 9oB 10oB 11oB

#1 1.0 -0.0235 0.3837 -0.0259 0.2127 -0.0371 0.0478 -0.0201 0.0010 -0.0090 0.0005 #2 0.8807 0.0643 0.0344 -0.1056 0.0183 0.0480 0.0150 -0.0101 0.0082 0.0001 -0.0103 #3 0.5500 -0.4966 0.0021 0.0213 -0.0600 -0.0403 0.0228 -0.0089 -0.0010 -0.0057 -0.0007 #4 0.0 -0.4966 0.0021 0.0213 -0.0600 -0.0403 0.0228 -0.0089 -0.0010 -0.0057 -0.0007

The welding induced residual stress distributions can be idealized to be composed

of tensile and compressive stress blocks, as shown in Fig.2.8. Along the welding line,

tensile (positive) residual stresses are usually developed with magnitude rtx in the x direction and rty in the y direction since welding is normally performed in both x

and y directions. In order to obtain equilibrium, corresponding compressive

(negative) residual stresses with magnitude rcx in the x direction and rcy in the y direction are developed in the middle part of plating. The breadths of the related

tensile residual stress blocks in the x and y directions can be shown to be as

follows:

-21-

rtxrcx

rcxt

bb2

= ,

rtyrcy

rcyt

a

a2

= (2.8)

where the tensile residual stress normally reaches the yield stress of material for mild

steel plating (e.g., ortyrtx = ), while it is usually somewhat less (approximately 80% of the material yield stress) for high tensile steel plating (e.g., ortyrtx 8.0 = ).

Once the magnitudes of the compressive and tensile residual stresses are known,

breadths of the tensile residual stress blocks can be determined from Eq.2.8. The

residual stress distributions in the x and y directions may be approximated by

-22-

rcxrcya

b = (2.11)

By substituting Eq.2.7 with Table 2.1 into Eq.2.6 or Eq.2.8, 2.10 and 2.11 into

Eq.2.9, the post-weld initial deflection and residual stress distribution can reasonably be

defined for practical design purposes.

2.5 Buckling Based Capacity

2.5.1 Design Equations

The basis of the plate capacity nominally adopted by most classification societies

is buckling. For one single stress component loading, the buckling based capacity B

(i.e., xB for xav , yB for yav and B for av ) is defined using the so-called Johnson-Ostenfeld equation (or sometimes called Bleich-Ostenfeld equation) to account for the effect of plasticity, as follows

( )

>

=

5.0if4

11

5.0if

k

E

E

kk

k

EE

B

(2.12)

where E = elastic buckling stress for one single stress component, (i.e., xE = as defined in Eq.2.26 for compressive xav , yE = as defined in Eq.2.27 for compressive

yav and E = as defined in Eq.2.28 for av ), ok = for either xav or yav , and 3/ook == for av . It is taken as oxB = for tensile xav and oyB = for

-23-

tensile yav .

The elastic buckling stress equations suggested in our study accommodate the in-

plane bending, lateral pressure, residual stress, and rotational restraints as necessary, but

the effect of initial deflection is not included since clear bifurcation buckling is not

defined for the initially deflected plating.

For combined stress component loading, the buckling based capacity component *

B is obtained as a solution of the following equations (comp.:-, tens.:+) (a) When both xav and yav are compressive:

12

B

av

2

yB

yav2

xB

xav=

+

+

(2.13a)

(b) When either xav , yav or both are tensile:

12

B

av

2

yB

yav

yB

yav

xB

xav

2

xB

xav=

+

+

(2.13b)

By taking xav as the reference (non-zero) stress component, for instance, the solution of Eq.2.13 with regard to xav is given by

When both xav and yav are compressive:

( )2yB

2xB

22

2B

2xB

21

2B

2yB

ByBxB*xB

CC1

++= (2.14a)

(b) When either xav , yav or both are tensile:

( )2yB

2xB

22

2B

2xB

21

2ByBxB1

2B

2yB

ByBxB*xB

CCCs

++= (2.14b)

-24-

where 1s = if xav is compressive and 1s = if xav is tensile. xav

yav1C

= ,

xav

av2C

= .

For safety evaluation using Eq.2.1, the buckling based capacity measure cB of

the plating under combined loading are therefore given by holding the loading ratio

constant, as follows

22

21

*

xBcB CC1 ++= (2.15)

A similar method can be applied to calculate the buckling based capacity measures

for the cases in which either yav or av is taken as the reference stress.

2.5.2 Validity of the Johnson-Ostenfeld Equation To account for the influence of plasticity on the buckling based capacity equation,

the Johnson-Ostenfeld formula is used, as defined in Eq.2.12. Figures 2.9 to 2.11 show

the validity of the Johnson-Ostenfeld equation by comparing with the nonlinear finite

element inelastic buckling (ultimate strength) solutions for the plating under one single stress component, varying the edge condition and the aspect ratio. In FEA, the

maximum initial deflaction of plating is t05.0w opl = and no welding residual stresses

are considered. It is seen that the Johnson-Ostenfeld equation generally has the

tendency to underestimate the inelastic buckling strength of the nearlly perfect plate

(Paik 1999).

-25-

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0xE/o

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

xu/

o

a/b = 3.0All edges remain straight (SE)

Johnson - Ostenfeld equation

: All edges simply supported (SS) : Simply supported alone longitudinal edges & clamped alone transverse edges (SC) : Clamped along longitudinal edges & simply supported along transverse edges (CS) : All edges clamped (CC)


under longitudinal compression alone, 0.3b/a = (symbol: FEA)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0yE/o

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

yu/

o





under transverse compression along, 3b/a = (symbol: FEA)

-26-

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0E/o

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

u/ o





under edge shear alone, 3b/a = (symbol: FEA)

2.5.3 Effect of Rotational Restraints The rotational restraints of the support members are included in the elastic

buckling equations for both xav and yav as parameters of influence. Ship plating is

supported by various types of members along the edges, which have a finite value of

the torsional rigidity. This is in contrast to the idealized simply supported boundary

conditions often assumed for design purposes. Depending on the torsional rigidity of

support members, the rotation along the plate edges will to some extent be restrained.

When the rotational restraints are zero, the edge condition corresponds to a simply

supported case, while the edge condition becomes clamped when the rotational

restraints are infinite.

Most current practical design guidelines from classification societies for the

buckling and ultimate strength of ship plating are based on boundary conditions in

which all (four) edges are simply supported. In real ship plating, idealized edge conditions such as simply supported or clamped however may never occur because of

finite rotational restraints.

-27-

According to the study of Paik et al. (1993) who investigated the bending and torsional rigidities of support members for deck, side and bottom plating in merchant

ships, the magnitude of the rotational restraint parameter L at long edges (ships longitudinal direction) is normally in the range of 0.05 to 3.0 (and usually not exceeding 5.0) while the amount S at the short edges (normal to the ship longitudinal direction) is normally in the range of 0.1 to 8.0 (and usually not exceeding 13.0). Thus, there is of course no case with zero or infinite rotational restraints in practice as long as

support members exist at their edges, and the amount of the rotational restraints at one

set of long or short edges is normally different from each other as well. It was also

found from the same investigation that the bending rigidities of support members are

usually sufficient enough so that the relative lateral deflection of typical members

providing the support to plating at edges can be taken to be small.

For advanced design of ship plating against buckling, it is hence important to

better understand the buckling strength characteristics of plating as a function of the

rotational restraints of support members along the edges. This problem has of course

been studied before, by a number of investigators. Lundquist & Stowell (1942) studied the effect of the edge condition on the buckling strength of rectangular plates subject to uniaxial compressive loads where the support along the unloaded edges was

intermediate between simply supported and clamped. Bleich (1952) and Timoshenko & Gere (1963) discussed the buckling strength of plates with various boundary conditions that one set of edges is elastically restrained while the other set of edges is either simply

supported or clamped. Gerard & Becker (1954) surveyed literature for the buckling of rectangular plates under various combinations of two or three types of loading under a

number of edge conditions. Evans (1960) carried out an extensive experimental study on the buckling strength of wide plates with the loaded (long) edges elastically restrained while the unloaded (short) edges are simply supported. Based on the experimental results, he derived a closed-form expression of the compressive strength

-28-

of wide plates taking into account the effect of rotational restraints along the loaded

edges. McKenzie (1963) studied the buckling strength of plating under biaxial compression, bending and edge shear that is simply supported along short edges (at which bending is applied) and elastically restrained along long edges.

These various previous studies are quite useful for the buckling strength design of

plating considering the rotational restraint effect along the edges. To the authors

knowledge, however, systematic investigations on the buckling strength of plating

which is elastically restrained along both long and short edges appear to be difficult to

come by and were thus needed. The aims of our study related to this issue (Paik & Thayamballi 2000) were to

investigate the buckling strength characteristics of plating with the boundary

conditions which are elastically restrained along the edges, and to

develop simple buckling design formulations of plating taking into account the

rotational restraints of support members along either one set of edges or all (four) edges.

The simplified formulations referred to are based on more exact solutions as

obtained by directly solving the buckling characteristic equations for a variety of the

torsional rigidities of support members and the plate aspect ratio. The characteristic

equation for the buckling of plating with elastic restraints along either long or short

edges while the other edges are simply supported is derived analytically. By solving the

characteristic equation, the buckling strength characteristics of plating are investigated

varying the plate aspect ratio and the torsional rigidity of support members. Based on

the computed results, closed-form expressions of the plate buckling strength are

obtained empirically by curve fitting. Simplified buckling design formulations for

plating with all edges elastically restrained are also derived.

Figures 2.12 to 2.15 show some selected sets of the buckling coefficients as

obtained by directly solving the theoretical characteristic buckling equation plotted

-29-

against the plate aspect ratio and the torsional rigidity of support members along the

plate edges (Paik & Thayamballi 2003). The accuracy of the proposed simplified equations obtained by curve fitting the more exact results may be verified by

comparison with the exact theoretical solutions, see Figs. 2.12 and 2.16 to 2.18. The

curve-fit design equations 1y2x1x k,k,k and 2yk are given in Appendix 1.

One of the useful insights developed herein is that the buckling coefficient for the

plating elastically restrained at both long and short edges can be expressed by a relevant

combination of the following three edge conditions, namely (a) elastically restrained at long edges and simply supported at short edges, (b) simply supported at long edges and elastically restrained at short edges, and (c) simply supported at all edges. Specifically it was noted that the following held approximately:

xo2x1xx kkkk += , yo2y1yy kkkk += (2.16)

where xk = buckling coefficient of plating elastically restrained at both long and short

edges for longitudinal compression, yk = buckling coefficient of plating elastically

restrained at both long and short edges for transverse compression, xok = buckling

coefficient of plating simply supported at all edges for longitudinal compression which

may be taken as 0.4k xo , and yok = buckling coefficient of plating simply supported

at all edges which may be taken as { }22yo )a/b(1k += . xok , 1xk , 2xk , yok , 1yk , 2yk = as defined in Appendix 1.

-30-

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

a/b

k x1

ExactApproximate

GJL/bD = 20.0GJL/bD = 2.0GJL/bD = 0.3GJL/bD = 0.0


elastically restrained at the long edges and simply supported at the short edges as

obtained by directly solving the buckling characteristic equation and by the proposed

approximate equation

0 1 2 3 4 5 63

4

5

6

7

8

9

k x2

a/b

GJS/aD = 20.0

GJS/aD = 0.4

GJS/aD = 0.2 GJS/aD = 0.1 GJS/aD = 0.0




-31-

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1

2

3

4

5

6

7

b/a

k y1

= 500.0= 20.0

= 10.0

= 4.0

= 2.0= 1.0

= 0.0

GJL/bD = oo

Fig.2.14. Buckling coefficient 1yk for a plate under transverse compression, elastically

restrained at the long edges and simply supported at the short edges as obtained by

directly solving the buckling characteristic equation

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1

2

3

4

5

6

7

8

k y2

b/a

= 10.0GJS/aD=oo

= 2.0= 1.0

= 0.5= 0.2

= 0.0

Fig.2.15. Buckling coefficient 2yk for a plate under transverse compression,



-32-

0 5 10 15 20 250

1

2

3

4

5

6

7

8

k x2

GJS/aD

ExactApproximate

a/b = 1.0

a/b = 1.5a/b = 2.0a/b = 3.0a/b = 5.0

Fig.2.16. Accuracy of the design equation for the buckling coefficient 2xk

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

GJL/bD

k y1

a/b = 1.0

a/b = 0.8

a/b = 0.5

a/b = 0.2

a/b = 0.0

ExactApproximate


-33-

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

k y2

GJS/aD

b/a = 1.0

= 0.9

= 0.8

= 0.5= 0.0

ExactApproximate


It was also found that the buckling interaction equation of the plating elastically

restrained along all edges and under combined loading can approximately take the same

relationship as that with simply supported conditions at all edges, but by replacing the

buckling stress components of the plating simply supported at all edges with the

corresponding ones for the elastically restrained plating. As two specific cases of

plating under combined biaxial compression or combined axial compression and edge

shear, where the plate edges are all clamped, i.e., with infinite rotational restraints,

Figures 2.19 and 2.20 show the elastic buckling interaction relations varying the plate

aspect ratio where the theoretical predictions were obtained by the formulae for simply

supported plates as given in Appendices 2 and 3 while FE solutions were calculated for

clamped plates.

-34-

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

FEM (ANSYS) : a/b=2: a/b=1

: a/b=3

a/b=1a/b=2

a/b=3

xE

xav

yE

yav

Fig.2.19. Elastic buckling interaction relationships for plating under combined biaxial

compression (symbol: eigen value finite element solutions for plating clamped at all edges, line: design equation for plating simply supported at all edges)

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

a/b=1a/b=2

a/b=3

FEM (ANSYS) : a/b=2: a/b=1

: a/b=3

E

av

xE

xav

Fig.2.20. Elastic buckling interaction relationships of plating under combined axial

compression and edge shear (symbol: eigen value finite element solutions for plating clamped at all edges, line: design equation for plating simply supported at all edges)

-35-

2.5.4 Effect of Residual Stresses The welding induced residual stress (compression: -, tension: +) is included in the

elastic buckling equations for both xav and yav as a parameter of influence. The

elastic buckling stress of simply supported plating under uniform axial compression

(i.e., without in-plane bending) is given by (Paik et al. 2000a, Paik & Thayamballi 2003) (For the symbols unless specified below, refer to the section on 2.4 modeling of fabrication related imperfections)

rey22

2

rex

2

2

2

xE bma

mba

a

mbtbD

+= (2.17)

where ( )

+=bb2

sin2bb

b2 t

trcxrtxrcxrex

,

( )

+=a

am2sin

m2a

aa

2 ttrcyrtyrcyrey

.

The second and third terms of the right hand side of Eq.2.17 reflect the effect of

welding induced residual stresses on the plate compressive buckling stress. m is the

buckling half wave number which is determined as a minimum integer satisfying the

following equation

rey22

22

2

2

bma

mba

a

mbtbD

+

+ ( ) rey2222

2

2

b1ma

b)1m(a

a

b)1m(tbD

++

++

+ (2.18)

Without the post-weld residual stresses, i.e., 0reyrex == , Eq.2.18 simplifies to

the well-known condition

-36-

( )1mmba ** + (2.19)

where *m is the buckling half wave number when the residual stresses do not exist. In the similar way, the elastic buckling stress yE of the simply supported plating

subject to axial compression in the y direction can be given by

reyrex2

22

2

2

2

2

yEa

ba

b1tbD

+= (2.20)

where rex and rey are defined as those in Eq.2.17 but replacing by 1m = . The

second and third terms of the right hand side of Eq.2.20 reflect the effect of welding

induced residual stresses.

Figure 2.21 shows the influence of welding induced residual stress on the

compressive buckling stress for the high tensile steel plating with the yield stress of

MPa352o = . In the calculations indicated in Fig.2.21, the level of residual stresses

and the plate slenderness ratio (i.e., t/b ratio) are varied. Two types of welding induced residual stresses in the y direction are presumed, namely one with zero

residual stresses and the other with rcxrcy a/b = . It is in the analysis assumed that

the magnitude of the tensile residual stresses is 80% of the yield stress, that is,

ortyrtx 8.0 == . It is evident from Fig.2.21 that the welding residual stresses can

significantly reduce the compressive buckling stress of the plating. The reduction

tendency of the buckling stress for thin plating is faster than that for thick plating, as

expected. It is also noted from Figs.2.21(c) and 2.21(d) that the residual stresses in the y direction may change the longitudinal buckling half wave number of the plating.

-37-

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.50.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.0005.0/ =orcx

15.0/ =orcx

30.0/ =orcx

50/ =tb0.0/ =orcy

*/

xExE

ba /

m/m* =1/1

m/m* =2/2

m/m* =3/3

m/m* =4/4

m/m* =5/5

Fig.2.21(a). Variation of the elastic compressive buckling stress (normalized by the elastic buckling compressive stress without residual stresses) as a function of the welding induced residual stress and the plate aspect ratio, 0rcy = , 50t/b = , 07.2= ,

MPa352o = , rtx o8.0 = ( *xE = buckling stress without residual stress)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.50.200.250.300.350.400.450.500.550.600.650.70

0.750.800.85

0.0/ =orcy 100/ =tb

*/

xExE

ba /

05.0/ =orcx

10.0/ =orcx

15.0/ =orcx

m/m* =1/1 m/m* =3/3 m/m* =5/5

m/m* =2/2 m/m* =4/4

Fig.2.21(b). Variation of the elastic compressive buckling stress (normalized by the elastic compressive buckling stress without residual stresses) as a function of the welding induced residual stress and the plate aspect ratio, 0rcy = , 100t/b = ,

14.4= , MPa352o = , rtx o8.0 = ( *xE = buckling stress without residual stress)

-38-

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

50/ =tb

rcxrcya

b =

05.0/ =orcx

15.0/ =orcx

30.0/ =orcx

*/

xExE

ba /

m/m*=1/1 m/m* =2/2 m/m*=3/3 m/m*=4/4 m/m* =5/5

m/m*=1/2 m/m* =2/3 m/m* =3/4 m/m* =4/5

Fig.2.21(c). Variation of the ela

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Ultimate Limit State Design of Ship Structures(1)